C₂×#126×C₃₇ #14557193

order	2368
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₃₇ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#126×C₃₇

Subgroups

	subgroup	G/H
center, socle	C₂³×C₃₇	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃₇
prime radical	C₂⁴×C₃₇	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃₇ ◁ G

246 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 55
transitive degree	μ_t(G) = 1184

A minimal faithful permutation representation — 4 generators on 55 points:

(1,2)(19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55);(4,8)(5,12)(10,17)(11,15)(13,16)(14,18);(3,4,9,15)(5,10,16,17)(6,8,7,11)(12,18,13,14);(3,5,6,13)(4,10,8,14)(7,12,9,16)(11,18,15,17)

Conjugacy classes & character table

1036 conjugacy classes. The character table is too large to display.

Relationships

maximal subgroup of	C₃₇×#1473, C₄×#126×C₃₇, C₃₇×#2334, …
maximal normal subgroup of	C₃₇×#1473, C₄×#126×C₃₇, C₃₇×#2334, …
maximal quotient of	C₂×C₃₇×#383, C₂×C₃₇×#387, C₂×C₃₇×#388, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(2368, 209);

C2×#126×C37 #14557193